Elliptic, parabolic and hyperbolic operators. Laplace, Laplace-Beltrami, Schrödinger, Dirac. Exterior derivative and Lie derivative operators.

learn more… | top users | synonyms

1
vote
1answer
294 views

Elliptic operators corresponds to non vanishing vector fields

Let $X$ be a non vanishing vector field on a compact manifold $M$. The only differential operator associated with $X$ which I am aware of, is the derivational operator $D(g)=X.g$. Unfortunately ...
13
votes
2answers
522 views

Codimension of the range of certain linear operators

Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map $D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the derivational operator correspond ...
6
votes
4answers
1k views

Symbol of pseudodiff operator

Hello, I am trying to understand the calculus of pseudodifferential operators on manifolds. All the textbooks I could put my hand on define the principal symbol of a pseudodifferential operator ...
14
votes
5answers
1k views

Book Recommendation - PDE's for geometricians / topologists

I am looking for recommendations for a book on partial differential equations, which is not written for applied mathematicians but rather focused on geometry and applications in topology, as well as ...
7
votes
2answers
748 views

What is the smallest $C^*$-algebra containing the “standard” pseudodifferential operators?

Is $\Psi^0(\mathbb{R})$ (pseudodifferential operators with symbols obeying $ |\partial^\alpha_x \partial^\beta_\xi a(x,\xi)| \leq C_{\alpha,\beta} (1+|\xi|)^{-|\beta|} $ ) a $C^*$-algebra? In other ...
14
votes
1answer
499 views

Derivation on real analytic manifolds

Let $M$ be a real analytic manifold. By $C^{\omega}(M)$ we mean the algebra of all analytic functions from $M$ to $\mathbb{R}$. Assume that $D$ is a derivation on $C^{\omega}(M)$ . Is there a ...
6
votes
0answers
211 views

Noncommutative geometry and line length

I would like to understand, in some formal sense, the relation between the Dirac operator and the line length introduced by Connes in noncommutative geometry. If $D$ is the Dirac operator, he sets $ds ...
5
votes
0answers
127 views

Does the normal ordered product on differential operators lift to $U\left(\mathfrak{gl}_n\right)$ ?

Let $n\in\mathbb N$. Let $k$ be a commutative ring. Let $\Omega$ denote the $k$-algebra of polynomial differential operators on $n$ variables $x_1$, $x_2$, ..., $x_n$ over $k$. (The multiplication in ...
5
votes
0answers
166 views

Differential operators that preserve real-rootedness

Is there some description of polynomial differential operators, $\mathcal{D}=\sum f_i(x) D_x^i$ such that, if $h$ is a polynomial all of whose roots are in $[0,1]$, then so are all the roots of ...
0
votes
0answers
298 views

A Generalized De Rham cohomology

Edit According to the comment of Alex Degtyarev, I deleted the last part of the previous version. Let $E$ be a real vector space. The complex valued $k$- tensors on $E$ is denoted by ...
1
vote
0answers
125 views

The “Rolle theorem” for sections of a vector bundle

1)Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that ...
-4
votes
1answer
249 views

Derivatives of infinite order [closed]

Is there any sense of taking an infinite number of derivatives? Is it discussed in the literature? For example, can one make sense of $$\frac{\partial^{\infty}f(x_1,x_2,\cdots)}{\partial x_1 ...